Your questions are scattered among multiple topics, so I will be focusing on the one in the title; for other questions please research them first ("what is parallel computing vs. parallel processing?").

I'm focusing on the questions: "Is there any parallel computing involved in scipy.linalg.solve?" and "Does it necessarily need all the matrice elements at once?".

Question 1: "Is there any parallel computing involved in scipy.linalg.solve?"

SciPy's linalg.solve itself does not directly handle parallelization, but it relies on optimized libraries for linear algebra operations like LAPACK (Linear Algebra PACKage), which can make use of parallelization internally (i.e. when running on multi-core processors). Whenever the installed libraries are compiled with parallelism or not depends on your system, of course, so the answer would depend on your installation.

For example, PLASMA is optimized for multi-core processors, since it is it's key feature.

Question 2: "Does it necessarily need all the matrice elements at once?".

When you use scipy.linalg.solve, you are solving the system Ax=b for x, and this function requires the matrix A and vector b as inputs. You need the entire matrix, yes.

If you have a sparse matrix, you should use scipy.sparse.linalg.spsolve instead, but if you need to solve for x or calculate the full inverse, SciPy expects access to all the elements of the matrix A at the start.